LINEAR FINITE ELEMENT APPROXIMATIONS FOR THE TIMOSHENKO BEAM AND THE SHALLOW ARCH PROBLEMS

1. Introductionln this paper we examine the linear finite element discretization of the Timoshenko beamand the shalIow arch problems. The thickness of beam and arch appears parametrically inthe model and locking phenomenon may occur due to the smaIl param…     

基于Maxwell模型的线性粘弹性波传导问题的半离散/全离散杂交应力有限元法

1引言设ΩСR²为一个凸的有界开集,边界为ЭΩ;T为一个正常数.我们考虑如下基于Maxwell模型的二维粘弹性固体介质波传导问题。     

非定常Stokes方程的全离散稳定化有限元格式

本文研究二维非定常Stokes方程全离散稳定化有限元方法.首先给出关于时间向后一步Euler半离散格式,然后直接从该时间半离散格式出发,构造基于两局部高斯积分的稳定化全离散有限元格式,其中空间用P_1—P_1元逼近,证明有限元解的误差估计.本文的研究方法使得理论证明变得更加简便,也是处理非定常Stokes方程的一种新的途径.     

A FULLY DISCRETE IMPLICIT-EXPLICIT FINITE ELEMENT METHOD FOR SOLVING THE FITZHUGH-NAGUMO MODEL

This work develops a fully discrete implicit-explicit finite element scheme for a parabolic-ordinary system with a nonlinear reaction term which is known as the FitzHugh-Nagumo model from physiology. The first-order backward Euler discretization for the time derivative, and an implicit-explicit discretization for the nonlinear reaction term are employed for the model, with a simple linearization technique used to make the process of solving equations more efficient. The stability and convergence of the fully discrete implicit-explicit finite element method are proved, which shows that the FitzHugh-Nagumo model is accurately solved and the trajectory of potential transmission is obtained. The numerical results are also reported to verify the convergence results and the stability of the proposed method.     

Sobolev方程的全离散有限体积元格式及数值模拟

本文研究二维Sobolev方程的有限体积元方法, 给出一种全离散化有限体积元格式及其有限体积元解的误差估计,并用数值例子说明数值计算的结果与理论结果是相吻合的, 进一步说明了有限体积元方法比其他数值方法更优越.     

粘弹性方程全离散化有限体积元格式及数值模拟

本文利用有限体积元方法研究二维粘弹性方程, 给出一种时间二阶精度的全离散化有限体积元格式, 并给出这种全离散化有限体积元解的误差估计, 最后用数值例子验证数值结果与理论结果是相吻合的. 通过与有限元方法和有限差分方法相比较, 进一步说明了全离散化有限体积元格式是求解二维粘弹性方程数值解的最有效方法之一.     

Sobolev方程的CN全离散化有限元格式

首先给出Sobolev方程关于时间二阶精度的Crank-Nicolson(CN)时间半离散格式,然后直接从时间二阶精度的CN时间半离散格式出发,构造CN全离散化的有限元格式,并给出这种时间二阶精度的CN全离散化有限元解的误差估计.本文研究方法使得理论证明变得更简便, 也是处理Sobolev方程的一种新的尝试.     

UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS

In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zerothorder term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H（div）-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.     

守恒混合元法在模拟污染物运移过程中的应用

研究污染物在土壤中运移的时空规律,为土壤环境质量评价及污染预测和防治提供科学的根据与途径,具有重要的理论和实际意义.通过建立土壤中污染物运移问题的全离散守恒混合元格式,讨论了守恒混合元解的存在唯一性,并给出了误差估计.最后给出了数值算例,数值模拟结果表明,用该方法模拟污染物运移问题是合理有效的.     

非定常Stokes方程的稳定化全离散有限体积元格式

本文研究非定常Stokes方程的有限体积元方法,给出一种基于两个局部高斯积分的稳定化全离散格式,并给其有限体积元解的误差分析.     

Stokes方程的稳定化间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过对通常的间断Galerkin有限元法应用稳定化思想,建立了一个相容的稳定间断有限元格式,对速度和压力的任意分片多项式空间Pl(K),Pm(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L2 范数的最优误差估计．     

COUPLING OF FINITE ELEMENT AND BOUNDARY ELEMENT METHODS FOR THE SCATTERING BY PERIODIC CHIRAL STRUCTURES

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R^3. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and nonhomogeneous. In this paper, variational formulations coupling finite element methods in the chiral medium with a method of integral equations on the periodic interfaces are studied. The well-posedness of the continuous and discretized problems is established. Uniform convergence for the coupling variational approximations of the model problem is obtained.     

WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS

The purpose of this paper is to discuss representations of high order C⁰finite element spaces on simplicial meshes in any dimension.When computing with high order piecewise polynomials the conditioning of the basis is likely to be important.The main result of this paper is a construction of representations by frames such that the associated L²condition number is bounded independently of the polynomial degree.To our knowledge,such a representation has not been presented earlier.The main tools we will use for the construction is the bubble transform,introduced previously in[1],and properties of Jacobi polynomials on simplexes in higher dimensions.We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.     

MIXED FINITE ELEMENT METHODS FOR A STRONGLY NONLINEAR PARABOLIC PROBLEM

1.IntroductionFOrsecondorderellipticproblems,themixedmethodwasdescribedandanalyzedbymanyauthors[1--3]inthecaseoflinearequationsindivergenceform,aswellasin[4,5]forquasilinearornonlinearproblemsindivergenceform.JohnsonandThorn6e[6]consideredalternativeproofSofthepreviouslyknownerrorestimatesforsuchmethodsintheellipticcase.Theyalsoanalyzedthemixedfiniteelementmethodforthepaxabolicequationgivenbypt--ho~f.Garcia[7]studiedtheconvergenceofmixedfiniteelementapproximationstoquasilinearparabolicequati…     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

LOW ORDER NONCONFORMING RECTANGULAR FINITE ELEMENT METHODS FOR DARCY-STOKES PROBLEMS

In this paper, we consider lower order rectangular finite element methods for the singularly perturbed Stokes problem. The model problem reduces to a linear Stokes problem when the perturbation parameter is large and degenerates to a mixed formulation of Poisson＇s equation as the perturbation parameter tends to zero. We propose two 2D and two 3D nonconforming rectangular finite elements, and derive robust discretization error estimates. Numerical experiments are carried out to verify the theoretical results.     

HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS

Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity.One key ingredient is the discrete reliability of a residualbased a posteriori error estimator,which controls the error of two discrete finite element solutions based on two nested triangulations.In the error analysis of nonconforming finite element methods,like the Crouzeix-Raviart or Morley finite element schemes,the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role and is the object of this paper.The nonconforming interpolation operator,which comes natural with the definition of the aforementioned nonconforming finite element in the sense of Ciarlet,allows for stability and approximation properties that enable direct proofs of the reliability for the residual that monitors the equilibrium condition.The novel approach of this paper is the suggestion of a right-inverse of this interpolation operator in conforming piecewise polynomials to design a nonconforming approximation of a given coarse-grid approximation on a refined triangulation.The results of this paper allow for simple proofs of the discrete reliability in any space dimension and multiply connected domains on general shape-regular triangulations beyond newest-vertex bisection of simplices.Particular attention is on optimal constants in some standard discrete estimates listed in the appendices.     

一类三维变动区域上的非线性抛物型方程之全离散有限元形式

1 引言 1986年,L.Cermak和M.Zlamal研究了半导体器件中杂质的重新分布,对具有活动边界的二维非线性扩散问题。给出在时间方向上是一阶精度的全离散有限元格式。证明了格式最优的H~1模和次最优的L~2模估计。1989年.P.Lesaint和R.Touzani对一维变动区域上的热传导方程。经过坐标变换,给出了在固定区域上的全离散有限元格式和最优的L~2模估计。1990年,梁国平和陈志明利用时空有限元,给出了变动区域上线性抛物型的方程的全离散变网格有限元格式。证明了最优的L~2收敛性。本文考虑了一类具有活动边界的三维